| L(s) = 1 | − 28.1·2-s + 3.02·3-s + 281.·4-s − 762.·5-s − 85.1·6-s − 5.57e3·7-s + 6.49e3·8-s − 1.96e4·9-s + 2.14e4·10-s + 4.76e4·11-s + 850.·12-s − 9.22e4·13-s + 1.56e5·14-s − 2.30e3·15-s − 3.27e5·16-s + 5.54e5·18-s − 8.37e3·19-s − 2.14e5·20-s − 1.68e4·21-s − 1.34e6·22-s − 3.64e5·23-s + 1.96e4·24-s − 1.37e6·25-s + 2.59e6·26-s − 1.19e5·27-s − 1.56e6·28-s + 3.50e6·29-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.0215·3-s + 0.549·4-s − 0.545·5-s − 0.0268·6-s − 0.877·7-s + 0.560·8-s − 0.999·9-s + 0.679·10-s + 0.981·11-s + 0.0118·12-s − 0.895·13-s + 1.09·14-s − 0.0117·15-s − 1.24·16-s + 1.24·18-s − 0.0147·19-s − 0.299·20-s − 0.0189·21-s − 1.22·22-s − 0.271·23-s + 0.0120·24-s − 0.702·25-s + 1.11·26-s − 0.0430·27-s − 0.481·28-s + 0.920·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 28.1T + 512T^{2} \) |
| 3 | \( 1 - 3.02T + 1.96e4T^{2} \) |
| 5 | \( 1 + 762.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.57e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.76e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.22e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 8.37e3T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.64e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.50e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.20e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.99e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 5.43e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.21e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.04e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.16e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.67e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.74e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.46e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.93e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.85e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.62e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.04e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.67e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.668720341109713187046560506547, −8.834655292933906516146475143405, −8.089174745145093893073124502028, −7.08470400376016431702491313171, −6.17131917365749526396101056156, −4.67215979427715568797818655297, −3.47024742771070516490288637762, −2.25750943059546731380225563786, −0.805958697511102051255867493180, 0,
0.805958697511102051255867493180, 2.25750943059546731380225563786, 3.47024742771070516490288637762, 4.67215979427715568797818655297, 6.17131917365749526396101056156, 7.08470400376016431702491313171, 8.089174745145093893073124502028, 8.834655292933906516146475143405, 9.668720341109713187046560506547
